Symmetric Geodesic Shape Averaging and Shape Interpolation

Abstract

Structural image registration is often achieved through diffeomorphic transformations. The formalism associated with the diffeomorphic framework allows one to define curved distances which are often more appropriate for morphological comparisons of anatomy. However, the correspondence problem as well as the metric distances across the database depend upon the chosen reference anatomy, requiring average transformations to be estimated. The goal of this paper is to develop an algorithm which, given a database of images, estimates an average shape based on the geodesic distances of curved, time-dependent transformations. Specifically, this paper will develop direct, efficient, symmetric methods for generating average anatomical shapes from diffeomorphic registration algorithms. The need for these types of averages is illustrated with synthetic examples and the novel algorithm is compared to the usual approach of averaging linear transformations. Furthermore, the same algorithm will be used for shape interpolation that is independent of the multi-scale framework used.